To find the best approximation of the solution, we first need to determine the ordered pair [tex]\((x, y)\)[/tex] where the values of [tex]\(y\)[/tex] from the two functions [tex]\(-x + 5\)[/tex] and [tex]\(6x - 1\)[/tex] are closest. We start by organizing our data and then finding the differences between the [tex]\(y\)[/tex]-values for various [tex]\(x\)[/tex]-values.
Step 1: Set up the table from the given data.
| [tex]\(x\)[/tex] | [tex]\(y = -x + 5\)[/tex] | [tex]\(y = 6x - 1\)[/tex] |
|---|---|---|
| 0.5 | 4.5 | 2 |
| 0.6 | 4.4 | 2.6 |
| 0.7 | 4.3 | 3.2 |
| 0.8 | 4.2 | 3.8 |
| 0.9 | 4.1 | 4.4 |
| 1.0 | 4 | 5 |
Step 2: Calculate the absolute differences [tex]\( | y_1 - y_2 | \)[/tex] for each pair [tex]\((x, y_1, y_2)\)[/tex].
[tex]\[
\begin{align*}
|4.5 - 2| & = 2.5 \\
|4.4 - 2.6| & = 1.8 \\
|4.3 - 3.2| & = 1.1 \\
|4.2 - 3.8| & = 0.4 \\
|4.1 - 4.4| & = 0.3 \\
|4 - 5| & = 1 \\
\end{align*}
\][/tex]
Step 3: Identify the smallest difference.
The smallest difference calculated is [tex]\(0.3\)[/tex]. This difference occurs for the [tex]\(x\)[/tex]-value [tex]\(0.9\)[/tex].
Step 4: Determine the ordered pair corresponding to the smallest difference.
At [tex]\(x = 0.9\)[/tex], the value of [tex]\(y\)[/tex] calculated from [tex]\(y = -x + 5\)[/tex] is [tex]\(4.1\)[/tex]. Therefore, the ordered pair is [tex]\((0.9, 4.1)\)[/tex].
Conclusion:
The best approximation of the solution to the system of equations [tex]\(y = -x + 5\)[/tex] and [tex]\(y = 6x - 1\)[/tex] within the given range is [tex]\((0.9, 4.1)\)[/tex].
